It seems to me that the question of the "truth" of the axiom of
infinity is almost entirely ridiculous, considering that the axiom is
such an extremely idealized version of real-world experience from which
the axiom is derived. And since the motivation for the question seems
ultimately to determine what is the value of the axiom, it seems to me
that the appropriate question is whether the axiom is *useful*. More
specifically, I mean whether the axiom is (ultimately) useful to
practicing scientists and engineers (and the like) interested in
solving "real-world" problems. By "ultimately" I mean that there is a
clear chain of usefulness, perhaps through intermediate mathematical
theories. Perhaps even more appropriate is the question: Does the
axiom of infinity make any essential difference to practicing
scientists and engineers (and the like) for the purpose of solving
"real-world" problems?
Since most basic concepts of mathematics, although often derived from
real-world experience, do not purport to represent that experience, it
seems that it is much more appropriate to ask whether the concepts are
useful rather than to ask whether they are true.
I favor the view that mathematics should be designed for solving
"real-world" problems.
Richard Haney
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